Is the Infinite an "identity" or part of reality, mainly being (the all or reality itself) or does the infinite lie but in our perceptions or consciousness, mainly being our imagination. I have an opinion, but before i say it i wish to demonstrate something i wrote not too long ago(reworded of course)

The Paradox of Movement by Aristotle,Aristotle was a guy who lived long ago who thought a lot about a lot of things and he came up with a brain teaser which went something like, if one was to shoot an arrow at a target, the arrow, before arriving at the target must first reach the half way point. but before it can reach the halfway point, it must reach the halfway point of the half, (or the quarter point) but before it can reach that, it must do reach the halfway point of that, and so on,, what you’re left with is an infinite amount of halfway points which the arrow must reach which leaves one with the conclusion that (however bizarre it may seem) all displacement is thus impossible, and for many years, no one has really seemed to be able to answer this riddle leaving people stupefied at this logic which seems to be incompatible with reality, yet seems too true.

But when one applies the concepts of relation(numbers) to things such as space, one finds that not everything is subject to this separate analysis since in the end the distinction which sets (this metre) from (that metre) is only mental and applicable only through us. The fact is that the world is not governed by human perceptions and that the Quanta seems to be the universal pixel, in which “half of it” cannot be. In the end numbers are not in them selves entities, but are only an a priori function of the mind to deal with relations to each other in terms of 1, 0 and infinity. Half, which the equal separation of 1, into 2, is logically, sound in comparative terms of macro space, but when one sees the true fabric of space, "1", looses all meaining as energy systems do not seem to follow such identities.

So just like the distance the arrow has to travel, mathematicians have long said that there can be infinite amount of points on a single line 1 cm long, though when applied to everything, infinity still holds its unatainability and encompasses all, and morelol a thought though for those who like to contemplate God, is God BEYOND INFINITY?

That would depend on how one defines God. I equate God with the Infinite.

Castilho12 wrote:if one was to shoot an arrow at a target, the arrow, before arriving at the target must first reach the half way point. but before it can reach the halfway point, it must reach the halfway point of the half, (or the quarter point) but before it can reach that, it must do reach the halfway point of that, and so on,, what you’re left with is an infinite amount of halfway points which the arrow must reach which leaves one with the conclusion that (however bizarre it may seem) all displacement is thus impossible

There is a point at which the arrow and the target become close enough that the target "reaches back" and the halfway point becomes a moot point. That particular half, infinity divided by two, does not exist, so it is skipped over.

Since everything is part of the greater whole anyway, it could also be viewed as the arrow didn't have any distance to travel in the first place because it already was there; was already part of the target to begin with. How can anything ever be halfway to where it already is?

As a side note, please excuse my painfully slow responses - I have internet connectivity issues.

However, the entire matter of discussion is moot considering exactly what is being done here Mathematically; the arrow and it's path to the target are being divided by an Infinite amount of 'half-way' points. What must be realized in this is that the distance which the arrow covers is Finite, and really it's all just a matter of nonanalogous terms: the Finite and the Infinite.

It is not hard to see how these are fundamentally uncompatible, as they are direct opposites. That which is Finite cannot be Infinite, and that which is Infinite cannot be Finite. Ergo, it is impossible to make the claim that you can divide a Finite quantity by an Infinite quantity. The reason the arrow hits the target in the end is because there isn't genuinely an infinite number of points in between them. You can count off the number of 'half-way' points given any two relative points, but that doesn't mean there is an Infinite number of them, simply because it gets to the point where there isn't a genuinely physical representation for the points to lie upon in space, at which point the midpoints made are symbolic, and nothing more.

That is to say, when you get down to such a scale where there is literally nothing but empty space between two atoms, quarks, what have you, there is nothing physical to represent the midpoints and they do not exist in Physical Reality, but only Metaphysically through the logical inference that given two points, a third exists as the midpoint. Of course, even given this understanding of these 'imaginary' points, the ultimate conclusion is that the distance covered by the arrow is not itself Infinite, and therefor cannot have a quality of any sort of being Infinite.

((As a side note, I would like to use this post, my first on this forum, as a greetings to all the minds out there. I don't often partake in forums, but I thought I'd give this one a shot. In short, hello everybody.))

It is not hard to see how these are fundamentally uncompatible, as they are direct opposites. That which is Finite cannot be Infinite, and that which is Infinite cannot be Finite.

Mathematicians do this all the time, think about limits or integrals were infinity can be the upper boundary of a finite entity like a surface. While in physics there are no infinite amounts of divisible lengths in a distance because of the minimum Planck length, one could make the same argument against any division to be actual and repeatable. Just that on larger scales one gets away by it because of the approximation of appearances, the uncertainties small enough not to influence the outcome noticeably.

That is to say, when you get down to such a scale where there is literally nothing but empty space between two atoms, quarks, what have you, there is nothing physical to represent the midpoints and they do not exist in Physical Reality, but only Metaphysically through the logical inference that given two points, a third exists as the midpoint.

All points are metaphysical when you put it that way, even those in your "physical reality". At the quantum level, "location" is a non-entity: just dynamic excitations of a quantum field, but that's still a mathematical description of it, not a metaphysical one.

Well Van Rhijn, there's two things I would like to point out about what I said concerning the Infinite and the Metaphysical:

Firstly, I subscribe to an Anaximander-esque view of Infinity, that similar to Apeiron. If you're not up to date on your Greek, it means "limitless, unbound", and with this concept, Anaximander theorized the creation of Reality through Infinite creation and Infinite destruction. In a more modern light, Apeiron is essentially an Infinite, non-repeating set which is the 'force', if you will, that makes all of Reality what it is.

Similarly, Infinity can also be seen as an endless multiplicity, akin to the infinite number of whole numbers: 1+1+1+1+1...ad infinitum. The difference between this Mathematical Infinity and Apeiron (or Apeironic Infinity) is that the first is cyclical, at least by the means of that you basically have an endless amount of a single unit, 1. 2 is two 1s, 3 is three 1s, 68 is sixty-eight 1s, and Infinity is infinite 1s, in this context. Apeiron is a set of all sets (semantics aside), and one set cannot be it's own set if it is identical to another.

With this in mind (which perhaps I should have noted in my first post, but eheu), what I meant by dividing a Finite quantity by an Infinite one is akin to a Limited quantity by an Unlimited one. The terms are still nonanalogous, and nearly paradoxical in combination.

As for the point about the Metaphysical points, I meant more along the lines of Subjective, only in one's head. In the example I gave, I said that there was nothing but empty space between the two points, so in fully Objective and Physical reality, there was nothing there to determine as a midpoint. However, with the Metaphysical logic of given two points, a midpoint exists between them, the Mind can invent a point of reference for the Midpoint. In the instance of a relatively larger scale, the arrow for example, there is genuinely particles that can be Objectively determined as being dead-center between them, though it is true enough that are a multitude of metaphysical points, as you so put it. This is simply a matter of the gap between the Subjective and Objective portions of reality.

Baron229 wrote:As for the point about the Metaphysical points, I meant more along the lines of Subjective, only in one's head. In the example I gave, I said that there was nothing but empty space between the two points, so in fully Objective and Physical reality, there was nothing there to determine as a midpoint. However, with the Metaphysical logic of given two points, a midpoint exists between them, the Mind can invent a point of reference for the Midpoint.

The problem with the paradox is that any "two points" are pure mathematical entities or perhaps even metaphysical but certainly not "physically" there somewhere, unless in a very roundabout generalized manner. Because of the limits of the mathematically "convenient" model used with normal counting and measuring, it goes wrong on different scales, like the ones you get by (near) infinitely dividing of multiplying entities.

In short: the points are abstract entities only meaningful within a certain predefined and assumed context. Like any other division really.

Baron229 wrote:Firstly, I subscribe to an Anaximander-esque view of Infinity, that similar to Apeiron. If you're not up to date on your Greek, it means "limitless, unbound", and with this concept, Anaximander theorized the creation of Reality through Infinite creation and Infinite destruction. In a more modern light, Apeiron is essentially an Infinite, non-repeating set which is the 'force', if you will, that makes all of Reality what it is.

That's all good and well but it remains tricky to talk about a set or attach other qualities or requirements to the infinite. Perhaps this is why Heraclitus said: "I see nothing other than becoming".

Diebert van Rhijn wrote:The problem with the paradox is that any "two points" are pure mathematical entities or perhaps even metaphysical but certainly not "physically" there somewhere, unless in a very roundabout generalized manner. Because of the limits of the mathematically "convenient" model used with normal counting and measuring, it goes wrong on different scales, like the ones you get by (near) infinitely dividing of multiplying entities.

The quantifications of Apeiron and Mathematical Infinity aside, which I could spend days elaborating on, I still think you misunderstand what I mean when I said the gap between the Physical and Metaphysical, Objective and Subjective.

Mathematically speaking or not, there is a path the arrow takes, or there is a set distance between two apples. What makes it divisible in the first place is that it exists as such, whether we measure it or not. Given this distance x, there are two methods of quantifying it: with established metric (i.e. the foot, inch, meter, kilometer, etc) or by totalling the objects between them, for the most precise measurements, atoms. Now, while the latter option is entirely absurd for practicality's sake, it is by far the more accurate because it does not deal with abstract units for measurement. Abstract units express distances of what is between two objects all the same, but the objects being used as metric are not physically there.

Going back to the Arrow: for practical purposes, we can say the arrow is 8 yards from the target when it is shot from the bow. It is common knowledge that a yard is 3 feet, a foot is 12 inches, an inch is 2.54 centimeters, et cetera, et cetera. The only thing is that these are abstact measurements; there are not literally atoms lined up into perfect rows into however many atoms inhabit a centimeter, merely for the fact it is basically impossible to do so. The legitimate distance between the arrow and the target is only represented by the jurisdiction of 8 yards; it does not physically represent what is objectively inbetween the two objects.

When I spoke of the finite distance of these two objects, I refered to this legitimate distance, not augmented by the abstract units of representative distance. As practical as they are, they are indeed, as you said, mathematical entities and can indeed be plugged into integrals and the like because that portion of mathematics is abstract itself.

Baron229 wrote:... established metric .. or .. totalling the objects between them, for the most precise measurements, atoms. Now, while the latter option is entirely absurd for practicality's sake, it is by far the more accurate because it does not deal with abstract units for measurement.

But you are still using abstract units for measurements in both cases. How do you think atoms are defined? Or quantum particles? Or how the meter is established by some wavelength. Distance or object size does not exist inherently, it's a relative measurement in the context of a model. A model which is defined for a certain scale or coordinate system only, which is why it can become untrustworthy when doing for example larger numbers (of intervals in this case).

What makes it divisible in the first place is that it exists as such, whether we measure it or not

It's divisible because we are measuring it using a scale which is part of a system which allows further division. It's always a representation, there is no "legitimate" distance as such. Just one which is accepted for practical purposes without defining in every distance the build-in limitations of the model. But that doesn't mean they are not there or the distance means anything outside the model.

Diebert van Rhijn wrote:But you are still using abstract units for measurements in both cases. How do you think atoms are defined? Or quantum particles? Or how the meter is established by some wavelength... It's always a representation, there is no "legitimate" distance as such.

It seems as if you are saying that the measurement came before the distance. The legitimate distance existed, it was measured according to the arbitary system, and that distance is represented through the measurement. Caesium 133 existed before the atomic clock, but the atomic clock is used over the atom because it's represetative distance (or time, in this case), is Metaphysically identical meaning that it can be used to, for most intents and purposes, accurately represent the atom itself. The only reason I used atoms as the "most precise measurement" is because atoms are the building blocks of matter, which we find handy to measure, on occasion.

Diebert van Rhijn wrote:Just one which is accepted for practical purposes without defining in every distance the build-in limitations of the model. But that doesn't mean they are not there or the distance means anything outside the model.

And I'm not saying that it isn't practical or the wrong way to do things. But in the context of this logic puzzle, the distances are Finite, Objective, and Physically manifest. They cannot exist (for the sake of what is being done Mathematically) Infinitely, Subjectively, or Metaphysically by simple logic. If "If A then B" is true, then " if ~A then ~B" is true, but "if ~A then B (or "if A then ~B")" is false.

((If you're not familiar with the modal operators here, I'd be happy to edit the post accordingly))

Baron229 wrote: The legitimate distance existed, it was measured according to the arbitary system, and that distance is represented through the measurement. Caesium 133 existed before the atomic clock...

But some arbitrary system always facilitated the initial measurement. Even a vague estimation like "this is a one day walk" invokes a certain definition invoking a lot of entities which will break down under extreme circumstances: walkers, weather, average speeds, state of the terrain and so on. What you name "legitimate distance" seems to be some estimate, based on some crude comparison with experiences from memory. Although very practical, its shortcomings and those of any other physical model are precisely the reason the paradox could appear in the first place. One could even muse the possibility that it's the unavoidable paradox of language and knowledge systems.

Diebert van Rhijn wrote: Although very practical, its shortcomings and those of any other psychical model are precisely the reason the paradox could appear in the first place. One could even muse the possibility it's the paradox of language and knowledge systems.

Well, this is what I said earlier: a gap between the Physical and Metaphysical, the Objective and Subjective.

In this context, Objective means that there is a distance traveled, whether we observe it or not, and it does not change based on our observations; it is not based per person. It is absolute in what it is.

Physical means that it is manifest outside of our Minds. That which is Metaphysical, conversely, is born from our Mind, our perception of perception. Bravery, for example, is Metaphysical. There is no Physical form for Bravery, nor any other Emotion or Abstract, and it is not quantifiable by the laws of Physics. Rather, it obeys the laws of Metaphysics: Logic and Reason. Bravery cannot be what Bravery is not; If B = ~A, then A =/= B. Mathematics is applied Logics, and in most cases Metaphysical, even though these logics are indeed applicable to the Physical world.

Under this reasoning, it genuinely isn't the semantics that is in question but the matter of perception. OUR perception. We have measured the distance to be x, and as such, x is associated under the seperate reasoning of our Mathematics, which are abstract, or Metaphysical, in this case (by calling it x it's even MORE Metaphysical!). The gap therein lies between our conceptualizations of x versus what x is meant to represent, and it represents a Finite quantity. In total, a Finite, Objective, and Physical quantity. The premise for the paradox is that, under the system of Mathematic Division for similar Abstract Quantities, there can be Infinite division of a Finite term. Again, I return to my original point of Nonanalogous Terms.

But when you say: "there is a distance traveled" as "absolute in what it is", I do wonder what you mean by "distance". The sensation of having traveled and the assumption because of changing scenery that the location has been changed? But a distance is a measurement using a scale. This is never "absolute" as a scale is relative and limited to a certain range. This is why I think the paradox can arise, since in the story of the paradox scale is implemented without known limitations. Perhaps this is what you mean, that the subjective is interpreted as objective? And by the way I prefer to drop the capitals, it's already confusing enough as it is.

Although we don't differ as much as it might seem, I still think your notion of "a Finite, Objective, and Physical quantity" seems spurious. That's just another abstract but more commonly applied. It might not be an abstract which goes very far when doing physics and its more exact modeling, but for daily usage such approximations and generalities just work until they break down into mysteries and paradoxes. They are certainly not more real or objective.

Diebert van Rhijn wrote: But a distance is a measurement using a scale. This is never "absolute" as a scale is relative and limited to a certain range. This is why I think the paradox can arise, since in the story of the paradox scale is implemented without known limitations. Perhaps this is what you mean, that the subjective is interpreted as objective?

You hit it on the head with the lattermost comment that the subjective is taken as objective fact, but to clarify what I mean by distance: there are two objects, X and Y, that have a genuine or representative number of objects (atoms, I suppose, just for consistancy's sake) between them. The number of objects between them is finite and this finity of objects, representative or not, is the distance between X and Y.

Diebert van Rhijn wrote: It might not be an abstract which goes very far when doing physics and its more exact modeling, but for daily usage such approximations and generalities just work until they break down into mysteries and paradoxes. They are certainly not more real or objective.

Well, that's where I disagree. Objective, physical reality trumps all notions of the subjective and metaphysical, merely for the fact that the subjective is never absolute and the metaphysical does not have an affect on the physical directly; that is to say, the laws of metaphysics do not effect physical things, nor do the laws of physics effect metaphysical things. We tend to deal with all of these in equal parts to the point where it is all clumped together in a simple conglomerate of reality, but when individuals take it to ponder such abstract notions as this or other paradoxes based on the characteristics of reality itself, we can't fall unto such a model as that proported by our "daily" system. Just like how light can act as a wave when we need it to or a ray at other times. We wouldn't be able to explain how the eye works if light was always a transverse squiggle, nor how color changes with wavelength with a ray.

Baron229 wrote:Objective, physical reality trumps all notions of the subjective and metaphysical, merely for the fact that the subjective is never absolute and the metaphysical does not have an affect on the physical directly; that is to say, the laws of metaphysics do not effect physical things, nor do the laws of physics effect metaphysical things.

But how can objective and physical reality trump the subjective and metaphysical if the very notion of having objectivity or physicality is already one particular powerful instance of applied metaphysics and necessarily subjective in terms of being subject to having a certain perspective? It's perhaps interesting to think of what's our actual reality principle. I think it's consistency, when a certain principle or finding is consistent with everything else we have found. It's about how things relate. The more they relate, the more real it can be called. This is where the scientific principle is getting close, a mathematical formula can be just as "real" as a dying star or a head ache simply because it can be related and be consistent with the remainder on many levels. A minimum of conflict and contradiction. This leads then to the ultimate metaphysical conclusion: the most real has to be the very thing which cannot be contradicted and remains always related to literary everything else in existence.